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The interval names shown in brackets could be said to be 'secondary', the others, 'primary'.
After releasing his system Keenan was informed that is was identical to the extended-diatonic interval-naming scheme of Adriaan Fokker but for the acknowledgment of more 11-limit ratios.This system depends on the tempering out of 81/80, where the diatonic major third, from four stacked fifths, approximates the just major third, 5/4. It also depends on the existence of neutral intervals, i.e., that the perfect fifth or equivalently, the chromatic semitone, subtends an even number of degrees of the ET. To simply to our familiar naming scheme for 12-tET, we observe that it applies to 24edo equally as directly as in 31-tET, where the prefixes correspond to degrees of the edo. Exactly the same is also true for 38-tET, twice 19-tET, a meantone which very closely approximates 1/3-comma meantone. Meantone temperament wherein the fifth is divided into two equally sized neutral thirds is referred to as neutral temperament. Whereas meantone temperament is generated by the fifth, iin neutral temperament the generator is half this interval, the neutral third. Where it was seen above that there are two neutral thirds, 9:11 and 22:27 that differ by 243/242, neutral temperament is at its most simple the temperament defined by this equivalence: the tempering out of 243/242, as meantone is defined by the tempering out of 81/80. The temperament that tempers out both 81/80 and 243/242 is called Mohajira, upon which Keenan's scheme can be said to be based.
The primary interval names resulting in this system's application to 24-tET and 38-tET is now show, along with 31-tET again for easy comparison, where 'M', 'm', 'P', 'N', 'A', 'd', 'S' and 's' are shorthand for major, minor, perfect, neutral, augmented, diminished, super and sub, respectively:
24edo: P1 S1/sm2 m2 N2 M2 SM2/sm3 m3 N3 M3 SM3/s4 P4 S4 A4/d5 s5 P5 S5/sm6 m6 N6 M6 SM6/sm7 m7 N7 M7 SM7/s8 P8
31edo: P1 S1 sm2 m2 N2 M2 SM2 sm3 m3 N3 M3 SM3 s4 P4 S4 A4 d5 s5 P5 S5 sm6 m6 N6 M6 SM6 sm7 m7 N7 M7 SM7 s8 P8
38edo: P1 S1 A1 sm2 m2 N2 M2 SM2 A2/d3 sm3 m3 N3 M3 SM3 d4 s4 P4 S4 A4 SA4/sd5 d5 s5 P5 S5 A5 sm6 m6 N6 M6 SM6 A6/d7 sm7 m7 N7 M7 SM7 d8 s8 P8
to Meantone tunings that are not Mohajira tunings, the regular diatonic interval names can be applied, but with the addition of double augmented and double diminished:
19edo: P1 A1 m2 M2 A2/d3 m3 M3 d4 P4 A4 d5 P5 A5 m6 M6 A6/d7 m7 d8 P8
26edo: P1 A2 d2 m2 M2 A2 d3 m3 M3 A3 d4 P4 A4 AA4/AA5 d5 P5 A5 d6 m6 M6 A6 d7 m7 M7 A7 d8 P8
Keenan adds further that if it is desired to distinguish between ratios that are in 31-tET approximated by the same number of steps, an addition prefix be added to describe the prime limit of the approximated interval. For 3-limit intervals, the obvious choice is 'Pythagorean', for 5-limit Keenan chooses 'classic', for 7, 'septimal, 11, 'undecimal' and 13, 'tridecimal'. When the highest prime is the same, Keenan suggests adding 'small' and 'large' as final prefixes for this purpose.
In non-Meantone tunings, the two definitions of major third - 4:5 and 64:81, the just (or classic) and Pythagorean major thirds no-longer correspond. If intervals are to receive unique names then to one or both of these major thirds must be added a prefix. Keenan has been involved with the development of both types of systems. Only when the major is defined by it's mapping as fourth fifths, i.e. 81/64, can conserve interval arithmetic.


=== [[Sagittal notation|Sagittal]] - [http://forum.sagittal.org/viewforum.php?f=9 sagispeak] ===
=== [[Sagittal notation|Sagittal]] - [http://forum.sagittal.org/viewforum.php?f=9 sagispeak] ===
One such system, sagispeak was developed by [[Dave Keenan]] and others as an interval naming system that maps 1-1 with the Sagittal microtonal music notation system. Sagittal notation was developed as a generalised diatonic-based notation system applicable equally to [[just intonation]], [[Equal Temperaments|equal tunings]] and rank-''n'' [[temperaments]]. Dozens of different accidentals can be used on a regular diatonic [[Staff notation|staff]] to notate up to extremely fine divisions, however in most cases only a handful are needed. In sagispeak, each accidental is presented by a prefix, made up of a single letter, in most cases, followed by either 'ai' if the accidental raises a note, or 'ao' if it lowers a note. Pythagorean intonation is assumed as a basis, like in previous microtonal ideas by Ellis, Helmholtz, Wolf, Monzo and others (combined into [http://tonalsoft.com/enc/h/hewm.aspx HEWM notation]). Then the prefixes depart from Pythaogrean intonation, altering by commas and introducing other primes. Noting that the set of 'sub' and 'super' 2nds, 3rds, 4ths and 5ths above all involve ratios of 7, Sagittal features an accidental of [[64/63]], the Archytas comma, which may be used to take a Pythagorean major third, or minor third to a supermajor third or subminor third respectively, for example. The prefix 'tao' takes Pythagorean minor 2nds, 3rds, 6ths and 7ths to subminor 2nds, 3rds, 6ths and 7ths respectively, and the prefix 'tai' takes Pythagorean major 2nds, 3rds, 6ths and 7ths to supermajor 2nds, 3rds, 6ths and 7ths respectively. Other prefixes include 'pai' and 'pao' (where 'p' is for 'pental', as in, involving prime 5), which raise or lower a note by [[81/80]], the meantone comma, respectively, and 'vai' and 'vao', which raise or lower a note by [[33/32]] respectively, leading to ratios of 11. The ratio [[5/4]], the just major third is considered almost universally to be a major third, however, so is the Pythagorean major third, 81/64. In [[Meantone]] temperament, with which we are very familiar, 81/80, the difference between these notes is tempered out, however in Sagittal, and arguably in any effective notation or interval naming system, these two major thirds need to be distinguished from each other. In Sagispeak 5/4 is a 'pao-major 3rd' (and 6/5 a 'pai-minor third), and 81/64, a major third. In meantone tunings they are both represented by the same interval, the major third.
One system in which 81/64 is the major third, sagispeak, was developed by [[Dave Keenan]] and others as an interval naming system that maps 1-1 with the Sagittal microtonal music notation system. Sagittal notation was developed as a generalised diatonic-based notation system applicable equally to [[just intonation]], [[Equal Temperaments|equal tunings]] and rank-''n'' [[temperaments]]. Dozens of different accidentals can be used on a regular diatonic [[Staff notation|staff]] to notate up to extremely fine divisions, however in most cases only a handful are needed. In sagispeak, each accidental is presented by a prefix, made up of a single letter, in most cases, followed by either 'ai' if the accidental raises a note, or 'ao' if it lowers a note. As in HEWM notation, Pythagorean intonation is assumed as a basis. Then the prefixes depart from Pythaogrean intonation, altering by commas and introducing other primes. In place of the prefixes 'sub' and 'super', generally signifying an alteration of 36/35 from 5-limit intervals or 64/63 for 3-limit, Sagittal features an accidental of [[64/63]], which may be used to take a Pythagorean major interval to a supermajor, minor to subminor, or perfect to super or sub. The prefix 'tao' indicates a decrease of 64/63 and and the prefix 'tai' an increase. Whereas in previous interval naming schemes 'major' and 'minor' were synonymous with the 5-limit tunings, in sagispeak they map instead to Pythagorean. A prefix is needed then to take a Pythagorean intoned interval to a 5-limit tuning. Where 5/4 is 81/80 below the the Pythagorean third, the prefixes 'pai' and 'pao' (where 'p' is for 'pental', as in, involving prime 5), which raise or lower a note by [[81/80]]<nowiki/>respectively. Similarly, 'vai' and 'vao', which raise or lower a note by [[33/32]] respectively, leading to ratios of 11.
 
Because it is built off of the diatonic scale, sagispeak conserves diatonic interval arithmetic, i.e. familiar relations in the diatonic scale, i.e. M2 + m3 = P4. As in Fokker/Keenan Extended-diatonic Interval-names, diatonic interval arithmetic is also extended, where, for example, tai-major 2 + tao-minor 3 = P4 (8/7 + 7/6 = 4/3), where opposite alterations cancel each other out, and diatonic interval arithmetic is conserved, a very useful property for a microtonal interval naming system to possess. Another helpful property of sagispeak is its generalised applicability to edos, just intonation and other tunings, where the same intervals maintain their spelling across different tunings. Despite these benefits however, many see Sagittal and Sagispeak as overly complex (even though the entire extended system need hardly ever be applied), and requiring too many new terms to be learnt.
 
We have seen that there are two competing definitions of a major third, the ratio '5/4' or the interval built from fourth stacked fifths, that may or may not correspond to 5/4. In meantone systems, those we are used to, they correspond, but in most edos they do not. Interval naming systems wherein the major third is defined as an approximation to 5/4 rather than as four fifths minus two octaves may benefit from a familiar name for 5/4, but they are unable to conserve diatonic interval arithmetic.


Because it is built off of the diatonic scale, sagispeak conserves diatonic interval arithmetic, i.e. familiar relations in the diatonic scale, i.e. M2 + m3 = P4. Diatonic interval arithmetic is also extended, where, for example, tai-major 2 + tao-minor 3 = P4 (8/7 + 7/6 = 4/3), where opposite alterations cancel each other out, and diatonic interval arithmetic is conserved . This is a very helpful property for a microtonal interval naming system to possess. Another helpful property of sagispeak is the generalised applicability to edos, just intonation and other tunings, where the same intervals maintain their spelling across different tunings. Despite these benefits however, many see Sagittal and Sagispeak as overly complex (even though the entire extended system need hardly ever be applied), and requiring too many new terms to be learnt.
For comparison, 31edo is shown below in sagispeak:


We have seen that there are two competing definitions of a major third, the ratio '5/4' or the interval built from fourth stacked fifths, that may or may not correspond to 5/4. Interval naming systems wherein the major third is defined as an approximation to 5/4 rather than as four fifths minus two octaves may benefit from a familiar name for 5/4, but they are unable to conserve diatonic interval arithmetic.
31edo: P1 tai-1/vai-1 tao-m2 m2 vai-2m/vao-M2 M2 tai-M2 tao-m3 m3 vai-m3/vao-M3 M3 tai-M3 tao-4 P4 vai-4 A4 d5 vao-5 P5 tai-5 tao-m6 m6 vai-m6/vao-M6 M6 tai-M6 tao-m7 m7 vai-m7.vao-M7 M7 tai-M7 vao-8/tao-8 P8


=== Dave Keenan's system ===
=== Dave Keenan's most recent system ===
In 2016 Dave Keenan proposed an alternative generalised [http://dkeenan.com/Music/EdoIntervalNames.pdf microtonal interval naming system for edos]. He uses the familiar prefixes 'sub', 'super', 'supra' and 'neutral'. His scheme is based on the diatonic scale, however the diatonic interval names are not defined by their position in a cycle of fifths like is Sagispeak. In Keenan's system the ET's best 3/2 is first labelled P5, and the fourth P4. The interval half-way between the tonic and fifth is labelled the neutral third, or 'N3', and halfway between the fourth and the octave N6. Then the interval a perfect fifth larger than N3 is labelled N7, and the interval a fifth smaller than N6 labelled N2. The neutral intervals then lie either at a step of the ET, or between two steps. After this the remaining interval names are decided based on the distance they lie in pitch from the 7 labelled intervals, which make up the ''Neutral scale'', P1 N2 N3 P4 P5 N6 N7, which, like the diatonic, is an MOS scale, which may be labelled [[Neutral7|Neutral[7]]] 3|3 using [[Modal UDP Notation|Modal UPD notation]]. To name an interval in an edo, the number of steps of [[72edo]] that most closely approximate the size of the interval difference from a note of the neutral scale is first found. Then the prefix corresponding to that number of steps of 72edo is applied to the interval name. The following chart details this process (can't load the chart :( ). An interval just smaller than a major third in Keenan's system is labelled a ''narrow major third'', and an interval just wider than a [[6/5]] minor third a ''wide minor third'', however he notes that 'narrow' and 'wide' are only necessary in edos greater than 31.  
In 2016 Dave Keenan proposed an alternative generalised [http://dkeenan.com/Music/EdoIntervalNames.pdf microtonal interval naming system for edos]. In what might be understood as a generalisation of his extended-diatonic interval-naming system described above onto any equal tuning. Employing as prefixes the familiar 'sub', 'super', and 'neutral'. His scheme is based on the diatonic scale, however the diatonic interval names are not defined by their position in a cycle of fifths like is Sagispeak. In Keenan's system the ET's best 3/2 is first labelled P5, and the fourth P4. The interval half-way between the tonic and fifth is labelled the neutral third, or 'N3', and halfway between the fourth and the octave N6. Then the interval a perfect fifth larger than N3 is labelled N7, and the interval a fifth smaller than N6 labelled N2. The neutral intervals then lie either at a step of the ET, or between two steps. After this the remaining interval names are decided based on the distance they lie in pitch from the 7 labelled intervals, which make up the ''Neutral scale'', P1 N2 N3 P4 P5 N6 N7, which, like the diatonic, is an MOS scale, which may be labelled [[Neutral7|Neutral[7]]] 3|3 using [[Modal UDP Notation|Modal UPD notation]]. To name an interval in an edo, the number of steps of [[72edo]] that most closely approximate the size of the interval difference from a note of the neutral scale is first found. Then the prefix corresponding to that number of steps of 72edo is applied to the interval name. The following chart details this process (can't load the chart :( ). An interval just smaller than a major third in Keenan's system is labelled a ''narrow major third'', and an interval just wider than a [[6/5]] minor third a ''wide minor third'', however he notes that 'narrow' and 'wide' are only necessary in edos greater than 31. Note that the interval just wide of a minor third is labelled a 'supraminor third' rather than a 'superminor third'. This reflect recent tendencies among microtonal musicians and theorists.  


Keenan's system is an elegant way to keep the 'major 3rd' label for 5/4, where labels depend on the size of the best fifth, however it suffers from it's applicability only to edos, and that it does not conserve interval arithmetic. Another potentially undesirable result of the system is that the major second approximates 10/9, and a ''wide major second'' 9/8, where as 9/8 is almost always considered a major second, and [[10/9]] often a narrow or small major second. One such system that considers 10/9 a narrow major second, is that of Aaron Hunt.  
Keenan's system is an elegant way to keep the 'major 3rd' label for 5/4, where labels depend on the size of the best fifth, however it suffers from it's applicability only to edos, and that it does not conserve interval arithmetic. Another potentially undesirable result of the system is that the major second approximates 10/9, and a ''wide major second'' 9/8, where as 9/8 is almost always considered a major second, and [[10/9]] often a narrow or small major second. One such system that considers 10/9 a narrow major second is that of Aaron Hunt.  


=== Size based systems ===
=== Size-based systems ===
Microtonal theorist [[Aaron Andrew Hunt]] devised [http://musictheory.zentral.zone/huntsystem4.html the Hunt system], which includes interval name assignments for JI (just intonation) and edos based on [[41edo]]. Compared to Keenan's 72 interval names, Aaron's system includes 41. His system is based directly on 41edo, and unlike Keenan's system, interval are given the name of the closest step of 41edo, and no account is taken of the size of the edos fifth. In 41edo, Major, minor, augmented and diminished intervals are those obtained through the approximately Pythagorean cycle of fifths. Intervals one step of 41edo above these are given the prefix 'small', one step larger are given the prefix 'large', two steps smaller the prefix 'narrow' and two larger the prefix 'wide'. As a result, 5/4 is labelled a 'small major 3rd', or SM3 (not to be confused with a super major third, a label that does not exist in this system).
Microtonal theorist [[Aaron Andrew Hunt]] devised [http://musictheory.zentral.zone/huntsystem4.html the Hunt system], which includes interval name assignments for JI (just intonation) and edos based on [[41edo]]. Compared to Keenan's 72 interval names, Aaron's system includes 41. His system is based directly on 41edo, and unlike Keenan's system, interval are given the name of the closest step of 41edo, and no account is taken of the size of the edos fifth. In 41edo, Major, minor, augmented and diminished intervals are those obtained through the approximately Pythagorean cycle of fifths. Intervals one step of 41edo above these are given the prefix 'small', one step larger are given the prefix 'large', two steps smaller the prefix 'narrow' and two larger the prefix 'wide'. As a result, 5/4 is labelled a 'small major 3rd', or SM3 (not to be confused with a super major third, a label that does not exist in this system).


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In Hunt's system when used in 41edo or JI diatonic interval arithmetic is conserved, but in other tunings it may not be, and Margo's system may not conserve diatonic interval arithmetic either. Both systems may be applied to arbitrary tunings, but the same intervals (defined, perhaps by a MOS scale) may not be given the same interval names across different tunings.
In Hunt's system when used in 41edo or JI diatonic interval arithmetic is conserved, but in other tunings it may not be, and Margo's system may not conserve diatonic interval arithmetic either. Both systems may be applied to arbitrary tunings, but the same intervals (defined, perhaps by a MOS scale) may not be given the same interval names across different tunings.


<ul><li>[[User:PiotrGrochowski/Extra-Diatonic Intervals]] gives each 43edo interval a name, then maps each desired interval to a 43edo interval.  [[User:PiotrGrochowski/Extra-Diatonic Intervals — 50edo]] does this with 50edo. '''It has a very amazingly excellent solution to the 5/4 and 81/64 problem''': The 5/4 is named major third and this notation is split in half, while 81/64 is named high major third. 10/9 and 9/8 are both named major second. [[User:PiotrGrochowski|PiotrGrochowski]] ([[Editor PiotrGrochowski|info]], [[User talk:PiotrGrochowski|talk]], [[Special:Contributions/PiotrGrochowski|contribs]])</li></ul>
<ul><li>[[User:PiotrGrochowski/Extra-Diatonic Intervals]] gives each 43edo interval a name, then maps each desired interval to a 43edo interval.  [[User:PiotrGrochowski/Extra-Diatonic Intervals — 50edo]] does this with 50edo. '''It has a very amazingly excellent solution to the 5/4 and 81/64 problem''': The 5/4 is named major third and this notation is split in half, while 81/64 is named high major third. 10/9 and 9/8 are both named major second. [[User:PiotrGrochowski|PiotrGrochowski]] ([[Editor PiotrGrochowski|info]], [[User talk:PiotrGrochowski|talk]], [[Special:Contributions/PiotrGrochowski|contribs]]). </li></ul>Size-based systems are completely generalisable, but do not conserve interval arithmetic.


=== Ups and Downs ===
=== Ups and Downs ===
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=== Primary well-ordered (unless otherwise noted) interval names for edos ===
=== Primary well-ordered (unless otherwise noted) interval names for edos ===
[[2edo]]: P1 P4/4-5/P5 P8
2edo: P1 P4/P5 P8


[[3edo]]: P1 3-4/P4 P5/5-6 P8  
3edo: P1 P4 P5 P8


[[4edo]]: P1 2-3 P4/4-5/P5 6-7 P8
4edo: P1 SM2/sm3 P4/P5 SM6/sm7 P8


5edo: P1/1-2 2-3 3-4/P4 P5/5-6 6-7 7-8/P8
5edo: P1/m2 M2/m3 M3/P4 P5/m6 M6/m7 M7/P8


6edo: P1 SM2 sm3/s4 P4/4-5/P5 S5/sM6 sm7 P8
6edo: P1 SM2 sm3/s4 P4/P5 S5/sM6 sm7 P8


7edo: P1 N2 N3 P4 P5 N6 N7 P8
7edo: P1 N2 N3 P4 P5 N6 N7 P8


8edo: P1 sM2 M2 P4 4-5 P5 m7 Sm7 P8
8edo: P1 sM2 M2 P4 S4/s5 P5 m7 Sm7 P8


9edo: P1 1-2 2-3 3-4 P4 P5 5-6 6-7 7-8 P8  
9edo: P1 M2 M3 m3 P4 P5 M6 m6 m7 P8


10edo: P1/1-2 N2 2-3 N3 3-4/P4 4-5 P5/5-6 N6 6-7 N7 7-8/P8
10edo: P1/m2 N2 M2/m3 N3 M3/P4 S4/s5 P5/m6 N6 M6/m7 N7 M7/P8


11edo: P1 Sm2 Sm3 N3 sM3 P4 P5 Sm6 N6 sM6 sM7 P8
11edo: P1 Sm2 Sm3 N3 sM3 P4 P5 Sm6 N6 sM6 sM7 P8


12edo: P1 m2 M2 m3 M3 P4 4-5 P5 m6 M6 m7 M7 P8
12edo: P1 m2 M2 m3 M3 P4 A4/d5 P5 m6 M6 m7 M7 P8


13edo: P1 N2 m3 M2 N3 P4 N5 N4 P5 N6 m7 M6 N7 P8
13edo: P1 N2 m3 M2 N3 P4 N5 N4 P5 N6 m7 M6 N7 P8


14edo: P1 1-2 N2 2-3 N3 3-4 P4 4-5 P5 5-6 N6 6-7 N7 7-8 P8
14edo: P1 S1/sm2 N2 SM2/sm3 N3 SM3/s4 P4 SA4/sd5 P5 S5/sm6 N6 SM6/sm7 N7 SM7/s8 P8


15edo: P1/1-2 S1/Sm2 sM2 2-3 Sm3 sM3 3-4/P4 S4 s5 P5/5-6 Sm6 sM6 6-7 Sm7 sM7/s8 7-8/P8
15edo: P1/m2 S1/Sm2 sM2 M2/m3 Sm3 sM3 M3/P4 S4 s5 P5/m6 Sm6 sM6 M6/m7 Sm7 sM7/s8 M7/P8


16edo: P1 1-2 Sm2 sM2 Sm3 sM3 3-4 P4 4-5 P5 5-6 Sm6 sM6 Sm7 sM7 7-8 P8
16edo: P1 1-2 Sm2 sM2 Sm3 sM3 3-4 P4 4-5 P5 5-6 Sm6 sM6 Sm7 sM7 7-8 P8
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17edo: P1 N1 N2 M2 m3 N3 M3 P4 N4 N5 P5 m6 N6 M6 m7 N7 N8 P8
17edo: P1 N1 N2 M2 m3 N3 M3 P4 N4 N5 P5 m6 N6 M6 m7 N7 N8 P8


19edo: P1 1-2 m2 M2 2-3 m3 M3 3-4 P4 A4 d5 P5 5-6 m6 M6 6-7 m7 M7 7-8 P8
19edo: P1 S1/sm2 m2 M2 SM2/sm3 m3 M3 SM3/s4 P4 A4 d5 P5 S5/sm6 m6 M6 SM6/sm7 m7 M7/s8 P8


21edo: P1 S1 sm2 N2 SM2 sm3 N3 SM3 s4 P4 SA4 sd5 P5 S5 sm6 N6 SM6 sm7 N7 SM7 d8 P8
21edo: P1 S1 sm2 N2 SM2 sm3 N3 SM3 s4 P4 SA4 sd5 P5 S5 sm6 N6 SM6 sm7 N7 SM7 d8 P8
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23edo: P1 S1 1-2 Sm2 sM2 2-3 Sm3 sM3 3-4 A4 P4 S4 s5 P5 d5 5-6 Sm6 sM6 6-7 Sm7 sM7 7-8 s8 P8 (can't quite get it well-ordered)
23edo: P1 S1 1-2 Sm2 sM2 2-3 Sm3 sM3 3-4 A4 P4 S4 s5 P5 d5 5-6 Sm6 sM6 6-7 Sm7 sM7 7-8 s8 P8 (can't quite get it well-ordered)


24edo: P1 1-2 m2 N2 M2 2-3 m3 N3 M3 3-4 P4 N4 4-5 N5 P5 5-6 m6 N6 M6 6-7 m7 N7 M7 7-8 P8
24edo: P1 S1/sm2 m2 N2 M2 SM2/sm3 m3 N3 M3 SM3/s4 P4 N4 A4/d5 N5 P5 S5/sm6 m6 N6 M6 SM6/sm7 m7 N7 M7 SM7/s8 P8


26edo: P1 A2 d2 m2 M2 A2 d3 m3 M3 A3 d4 P4 A4 4-5 d5 P5 A5 d6 m6 M6 A6 d7 m7 M7 A7 d8 P8
26edo: P1 A2 d2 m2 M2 A2 d3 m3 M3 A3 d4 P4 A4 4-5 d5 P5 A5 d6 m6 M6 A6 d7 m7 M7 A7 d8 P8
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28edo: P1 sA1 1-2 Sm2 N2 sM2 2-3 Sm3 N3 sM3 3-4 S4 N4 sA4 4-5 Sd5 N5 s5 5-6 Sm6 N6 sM6 6-7 Sm7 N7 sM7 7-8 Sd8 P8
28edo: P1 sA1 1-2 Sm2 N2 sM2 2-3 Sm3 N3 sM3 3-4 S4 N4 sA4 4-5 Sd5 N5 s5 5-6 Sm6 N6 sM6 6-7 Sm7 N7 sM7 7-8 Sd8 P8


29edo: P1 1-2 m2 Sm2 sM2 M2 2-3 m3 Sm3 sM3 M3 3-4 P4 S4 d5 A4 s5 P5 5-6 m6 Sm6 sM6 M6 6-7 m7 Sm7 sM7 M7 7-8 P8
29edo: P1 S1 m2 Sm2 sM2 M2 SM2/sm3 m3 Sm3 sM3 M3 SM3/s4 P4 S4 d5 A4 s5 P5 S5/sm6 m6 Sm6 sM6 M6 SM6/sm7 m7 Sm7 sM7 M7 s8 P8


31edo: P1 N1 sm2 m2 N2 M2 SM2 sm3 m3 N3 M3 SM3 s4 P4 N4 A4 d5 N4 P5 S5 sm6 m6 N6 M6 SM6 sm7 m7 N7 M7 SM7 N8 P8
31edo: P1 N1 sm2 m2 N2 M2 SM2 sm3 m3 N3 M3 SM3 s4 P4 N4 A4 d5 N4 P5 S5 sm6 m6 N6 M6 SM6 sm7 m7 N7 M7 SM7 N8 P8


34edo: P1 1-2 m2 Sm2 N2 sM2 M2 2-3 m3 Sm3 N3 sM3 M3 3-4 P4 S4 N4 4-5 N5 s5 P5 5-6 m6 Sm6 N6 sM6 M6 6-7 m7 Sm7 N7 sM7 M7 7-8 P8
34edo: P1 S1 m2 Sm2 N2 sM2 M2 2-3 m3 Sm3 N3 sM3 M3 3-4 P4 S4 N4 4-5 N5 s5 P5 5-6 m6 Sm6 N6 sM6 M6 6-7 m7 Sm7 N7 sM7 M7 s8 P8  


38edo: P1 S1 1-2 sm2 m2 N2 M2 SM2 2-3 sm3 m3 N3 M3 SM3 3-4 s4 P4 N4 A4 4-5 d5 N5 P5 S5 5-6 sm6 m6 N6 M6 SM6 6-7 sm7 m7 N7 M7 SM7 7-8 s8 P8
38edo: P1 S1 A1 sm2 m2 N2 M2 SM2 A2/d3 sm3 m3 N3 M3 SM3 d4 s4 P4 N4 A4 SA4/sd5 d5 N5 P5 S5 A5 sm6 m6 N6 M6 SM6 A6/d7 sm7 m7 N7 M7 SM7 d8 s8 P8


41edo: P1 N1 sm2 m2 Sm2 N2 sM2 M2 SM2 sm3 m3 Sm3 N3 sM3 M3 SM3 s4 P4 S4 N4 sA4 Sd5 N5 sM5 P5 S5 sm6 m6 Sm6 N6 sM6 M6 SM6 sm7 m7 Sm7 N7 sM7 M7 SM7 N8 P8
41edo: P1 N1 sm2 m2 Sm2 N2 sM2 M2 SM2 sm3 m3 Sm3 N3 sM3 M3 SM3 s4 P4 S4 N4 sA4 Sd5 N5 sM5 P5 S5 sm6 m6 Sm6 N6 sM6 M6 SM6 sm7 m7 Sm7 N7 sM7 M7 SM7 N8 P8
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